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4.3 Multi-label classification with high dimensionality This study mainly focuses on the multi-dimensional output of [MATH] . This is because the number of labels is much smaller than the number of hidden units in most practice cases. However, since classification problems with far more labels are sometimes examined in... |
, it would be helpful to remark on the case of [MATH] here. Denote the mean of the FIM’s eigenvalues in the case of [MATH] as [MATH] and so on. Straightforwardly, we can derive |
[EQUATION] [EQUATION] The derivation is shown in Supplementary Material A.5. The mean of eigenvalues has the same form as Eq. ( 13 ) obtained in the case of [MATH] . The second moment and maximum eigenvalues can be evaluated by the form of inequalities. We found that the mean is of [MATH] while the maximum eigenvalue i... |
Conclusion and discussion The present work elucidated the asymptotic statistics of the Fisher information matrix (FIM) common among deep networks with any number of layers and various activation functions. The statistics of FIM are characterized by the small mean of eigenvalues and the huge maximum eigenvalue, which ar... |
We demonstrated that the experiments with the Gaussian prior on the parameters coincided well with the theory. Basically, the mean field theory is based on the central limit theorem with the parameters generated in an i.i.d. manner with finite variances. Therefore, one can expect that the good agreement with the theory... |
The derived statistics are also of potential importance to other learning strategies, for instance, natural gradient methods. When the loss landscape is non-uniformly distorted, naive gradient methods are likely to diverge or become trapped in plateau regions, but the natural gradient, [MATH] , converges more efficient... |
. Because it normalizes the distortion of the loss landscape, the naive extension of Section 4.2 to the natural gradient leads to [MATH] and it seems to be much easier to choose the appropriately sized learning rate. However, we found that the FIM has many eigenvalues close to zero, and the inversion of it would make t... |
. The development of practical natural gradient methods will require modification such as damping. It would also be interesting for our framework to quantitatively reveal the effects of normalization methods on the FIM. In particular, batch normalization may alleviate the larger eigenvalues because it empirically allow... |
It would also be fruitful to investigate the eigenvalues of the Hessian with a large error ( ) and to theoretically quantify the negative eigenvalues that lead to the existence of saddle points and the loss landscapes without spurious local minima |
. The global structure of the parameter space should be also explored. We can hypothesize that the parameters are globally connected through the locally flat dimensions and compose manifolds of flat minima. |
Our framework on FIMs is readily applicable to other architectures such as convolutional networks and residual networks by using the corresponding mean field theories |
To this end, it may be helpful to remark that macroscopic variables in residual networks essentially diverge at the extreme depths |
If one considers extremely deep residual networks, the statistics will require a careful examination of the order of the network width and the explosion of the macroscopic variables. We expect that further studies will establish a mathematical foundation of deep learning from the perspective of the large limit. |
Acknowledgments This work was partially supported by a Grant-in-Aid for Research Activity Start-up (17H07390) from the Japan Society for the Promotion of Science (JSPS). |
Supplementary Materials Appendix A Proofs A.1 Theorem 1 (i) Case of [MATH] To avoid complicating the notation, we first consider the case of the single output ( [MATH] ). The general case is shown after. The network output is denoted by [MATH] here. We denote the Fisher information matrix with full components as |
[EQUATION] where we notice that [EQUATION] In general, the sum over the eigenvalues is given by the matrix trace, [MATH] We denote the average of the eigenvalues of the diagonal block as [MATH] for [MATH] , and [MATH] for [MATH] . Accordingly, we find |
[EQUATION] The contribution of [MATH] is negligible in the large [MATH] limit as follows. The first term is [EQUATION] We can apply the central limit theorem to summations over the units [MATH] and [MATH] independently because they do not share the index of the summation. By taking the limit of [MATH] , we obtain |
[MATH] . The variable [MATH] is computed by the recursive relation (9). Under the Assumption 1, [MATH] is given by the recursive relation (11). Note that this transformation to the macroscopic variables holds regardless of the sample index [MATH] . Therefore, we obtain |
[EQUATION] where [MATH] comes from [MATH] , and [MATH] comes from [MATH] In contrast, the contributions of the bias entries are smaller than those of the weight entries in the limit of [MATH] , as is easily confirmed: |
[EQUATION] [MATH] is [MATH] while [MATH] is [MATH] . Hence, the mean [MATH] is negligible and we obtain [MATH] (ii) [MATH] of [MATH] |
We can apply the above computation of [MATH] to each network output [MATH] [MATH] ): [EQUATION] Therefore, the mean of the eigenvalues becomes |
[EQUATION] [MATH] A.2 Corollary 2 Because the FIM is a positive semi-definite matrix, its eigenvalues are non-negative. For a constant [MATH] , we obtain |
[EQUATION] This is known as Markov’s inequality. When [MATH] , combining this with Theorem 1 immediately yields [EQUATION] Because [MATH] is also a trivial upper bound of [MATH] , we obtain Corollary 2. |
[MATH] A.3 Theorem 3 Let us describe the outline of the proof. One can express the FIM as [MATH] by definition. Here, let us consider a dual matrix of [MATH] , that is, [MATH] [MATH] and [MATH] have the same nonzero eigenvalues. Because the sum of squared eigenvalues is equal to [MATH] , we have [MATH] . The non-diagon... |
[MATH] corresponds to an inner product of the network activities for different inputs [MATH] and [MATH] , that is, [MATH] The diagonal entry [MATH] is given by [MATH] Taking the summation of [MATH] over all of [MATH] and [MATH] , we obtain the theorem. In particular, when [MATH] and [MATH] [MATH] is equal to the square... |
The detailed proof is given as follows. (i) Case of [MATH] Here, let us express the FIM as [MATH] , where [MATH] is a [MATH] matrix whose columns are the gradients on each input sample, i.e., [MATH] |
[MATH] We also introduce a dual matrix of [MATH] , that is, [MATH] [EQUATION] Note that [MATH] is a [MATH] matrix while [MATH] is a [MATH] matrix. We can easily confirm that these [MATH] and [MATH] have the same non-zero eigenvalues. |
The squared sum of the eigenvalues is given by [MATH] . By using the Frobenius norm [MATH] , this is [MATH] Similar to [MATH] , the bias entries in [MATH] are negligible because the number of the entries is much less than that of weight entries. Therefore, we only need to consider the weight entries. The [MATH] -th ent... |
[EQUATION] where we defined [EQUATION] We can apply the central limit theorem to [MATH] and [MATH] independently because they do not share the index of the summation. For [MATH] , we have [MATH] and [MATH] in the limit of [MATH] , where the macroscopic variables [MATH] and [MATH] satisfy the recurrence relations (10) a... |
[MATH] and [MATH] are constants of [MATH] Then, for all [MATH] and [MATH] [EQUATION] Similarly, for [MATH] , we have [MATH] [MATH] and then [MATH] |
Thus, under the limit of [MATH] , the dual matrix is asymptotically given by [EQUATION] Neglecting the lower order term, we obtain |
[EQUATION] Note that, when [MATH] [MATH] becomes zero and the lower order term may be non-negligible. In this exceptional case, we have [MATH] , where the second term comes from the [MATH] term of Eq. (A.23). Therefore, the lower order evaluation depends on the [MATH] ratio, although it is outside the scope of this stu... |
(ii) [MATH] of [MATH] Here, we introduce the following dual matrix [MATH] [EQUATION] where [MATH] is a [MATH] matrix whose columns are the gradients on each input sample, i.e., [MATH] |
[MATH] and [MATH] is a [MATH] matrix. The FIM is represented by [MATH] [MATH] is a [MATH] matrix and consists of [MATH] block matrices, |
[EQUATION] for [MATH] The diagonal block [MATH] is evaluated in the same way as the case of [MATH] . It becomes [MATH] as shown in Eq. (A.23). The non-diagonal block [MATH] has the following [MATH] -th entries: |
[EQUATION] Under the limit of [MATH] , while [MATH] becomes [MATH] of [MATH] [MATH] becomes zero and its lower order term of [MATH] appears. This is because the different outputs ( [MATH] ) do not share the weights [MATH] . We have [MATH] and then obtain [MATH] [MATH] ) through the backpropagated chain (7). Thus, the e... |
[EQUATION] where [MATH] is the Kronecker delta. After all, we have [EQUATION] where the first term comes from the diagonal blocks of [MATH] and the second one is their lower order term. The third term comes from the non-diagonal blocks of [MATH] As one can see from here, when [MATH] , the thrid term becomes non-negligi... |
A.4 Theorem 4 (i) Case of [MATH] Because [MATH] and [MATH] have the same non-zero eigenvalues, what we should derive here is the maximum eigenvalue of [MATH] . As shown in Eq. (A.23), the leading term of [MATH] asymptotically becomes [MATH] in the limit of [MATH] The eigenvalues of [MATH] are explicitly obtained as fol... |
[MATH] (ii) [MATH] of [MATH] Let us denote [MATH] shown in Eq. (A.31) by [MATH] [MATH] is the leading term of [MATH] and given by a [MATH] block diagonal matrix whose diagonal blocks are given by [MATH] [MATH] denotes the residual term of [MATH] In general, the maximum eigenvalue is denoted by the spectral norm [MATH] ... |
[EQUATION] We can obtain [MATH] because the maximum eigenvalues of the diagonal blocks are the same as the case of [MATH] . Regarding [MATH] this is bounded by [MATH] . Therefore, when [MATH] , we can neglect [MATH] of [MATH] compared to [MATH] of [MATH] |
On the other hand, we can also derive the lower bound of [MATH] as follows. In general, we have [EQUATION] Then, we find [EQUATION] |
where [MATH] is a [MATH] -dimensional vector whose first [MATH] entries are [MATH] and the others are [MATH] , that is, [MATH] We can compute this lower bound by taking the sum over the entries of [MATH] , which is equal to Eq. (A.23): |
[EQUATION] Finally, we find that the upper bound (A.34) and lower bound (A.37) asymptotically take the same value of [MATH] , that is, |
[MATH] [MATH] A.5 Case of [MATH] The mean of eigenvalues [MATH] is derived in the same way as shown in Section A.1 (ii), that is, [MATH] |
Regarding the second moment [MATH] , the lower order term becomes non-negligible as remarked in Eq. (A.33). We evaluate this [MATH] by using inequalities as follows: |
[EQUATION] As shown in Section A.3, for any [MATH] , we obtain [MATH] in the limit of [MATH] Thus, the lower bound becomes the same form as [MATH] , That is, [MATH] In contrast, the upper bound is given by |
[EQUATION] where [MATH] denotes the FIM of the [MATH] -th output, i.e., [MATH] . Therefore, the upper bound is reduced to the summation over [MATH] of [MATH] . In the limit of [MATH] , we obtain [MATH] |
Next, we show inequalities for [MATH] . We have already derived the lower bound (A.37) and this bound holds in the case of [MATH] as well. In contrast, the upper bound (A.34) may become loose when [MATH] is larger than [MATH] because of the residual term [MATH] . Although it is hard to explicitly obtain the value of [M... |
[MATH] A.6 Theorem 5 The Fisher-Rao norm is written as [EQUATION] where [MATH] represents an entry of the FIM, that is, [MATH] . Because [MATH] includes the random variables [MATH] and [MATH] , we consider the following expansion. Note that [MATH] and [MATH] are infinitesimals [MATH] Performing a Taylor expansion aroun... |
[EQUATION] where [MATH] is the parameter set [MATH] with [MATH] . By substituting the above expansion into the Fisher-Rao norm and taking the average [MATH] , we obtain the following leading term: |
[EQUATION] For, [MATH] , the last line becomes zero because of [MATH] . For [MATH] , we have [MATH] . After all, in the limit of [MATH] , we obtain |
[EQUATION] where the derivation of the macroscopic variables is similar to that of [MATH] , as shown in Section A.1. Since we have [MATH] , it is easy to confirm [MATH] . When all [MATH] take the same value, we have [MATH] and the equality holds. |
[MATH] A.7 Lemma 6 Suppose a perturbation around the global minimum: [MATH] Then, the gradient update becomes [EQUATION] where we have used [MATH] and [MATH] |
Consider a coordinate transformation from [MATH] to [MATH] that diagonalizes [MATH] . It does not change the stability of the gradients. Accordingly, we can update the [MATH] -th component as follows: |
[EQUATION] Solving its characteristic equation, we obtain the general solution, [EQUATION] where [MATH] and [MATH] are constants. This recurrence relation converges if and only if [MATH] for all [MATH] . Therefore, |
[MATH] is necessary for the steepest gradient to converge to [MATH] [MATH] Appendix B Analytical recurrence relations B.1 Erf networks |
Consider the following error function as an activation function [MATH] [EQUATION] The error function well approximates the tanh function and has a sigmoid-like shape. For a network with [MATH] , the recurrence relations for macroscopic variables do not require numerical integrations. |
(i) [MATH] and [MATH] Note that we can analytically integrate the error functions over a Gaussian distribution: [EQUATION] Hence, the recurrence relations for the feedforward signals (9) have the following analytical forms: |
[EQUATION] Because the derivative of the error function is Gaussian, we can also easily integrate [MATH] over the Gaussian distribution and obtain the following analytical representations of the recurrence relations (11): |
[EQUATION] (ii) [MATH] and [MATH] To compute the recurrence relations for the feedforward correlations (10), note that we can generally transform [MATH] into |
[EQUATION] For the error function, [EQUATION] and we obtain [EQUATION] This is the analytical form of the recurrence relation for [MATH] |
Finally, because the derivative of the error function is Gaussian, we can also easily obtain [EQUATION] This is the analytical forms of the recurrence relations for [MATH] |
B.2 ReLU networks We define a ReLU activation as [MATH] For a network with this ReLU activation function, the recurrence relations for the macroscopic variables require no numerical integrations. |
(i) [MATH] and [MATH] We can explicitly perform the integrations in the recurrence relations (9) and (11): [EQUATION] (ii) [MATH] and [MATH] |
We can explicitly perform the integrations in the recurrence relations (10) and (12): [EQUATION] where [MATH] B.3 Linear networks |
We define a linear activation as [MATH] For a network with this linear activation function, the recurrence relations for the macroscopic variables do not require numerical integrations. |
We can explicitly perform the integrations in the recurrence relations (10) and (12): [EQUATION] Appendix C Additional Experiments C.1 Dependence on [MATH] C.2 Training on CIFAR-10 |
# Source: arxiv 1806.01322 # Title: Past Visions of Artificial Futures: One Hundred and Fifty Years under the Spectre of Evolving Machines # Sections: all # Downloaded: 2026-03-03T01:55:05.670323+00:00 |
Past Visions of Artificial Futures One Hundred and Fifty Years under the Spectre of Evolving Machines Abstract The influence of Artificial Intelligence (AI) and Artificial Life (ALife) technologies upon society, and their potential to fundamentally shape the future evolution of humankind, are topics very much at the fo... |
Introduction “And why should one say that the machine does not live? It breathes … It moves … And has it not a voice? … And yet the mystery of mysteries is to view machines making machines; a spectacle that fills the mind with curious, and even awful, speculation.” |
Coningsby (Disraeli,, 1844 , p. 154) By the climax of the British Industrial Revolution in the early 1800s, the widespread introduction of increasingly sophisticated manufacturing machines had raised anxiety about the potential long-term consequences of mechanisation. Areas of unease included not just the impact of tec... |
(Archer,, 2000 , but also the growing appreciation of the self-amplifying potential of the new machines. In 1844, the British author and future prime minister Benjamin Disraeli wrote the novel Coningsby . In a section describing the industrial landscape of Manchester, the narrator raises the idea of machines making mac... |
During the same period, the scientific understanding of the complexity of biological life was undergoing a revolution, in the theories being developed by Charles Darwin and Alfred Russell Wallace. Both theories were first presented at the Linnean Society of London in 1858 (Darwin and Wallace,, 1858 , with a greatly ext... |
(Darwin,, 1859 At this time, the intellectual elite of England were a richly connected web of thinkers, among whom ideas of science, philosophy, technology, literature and the arts freely flowed. It did not take long for the contemporaneous ideas of machines making machines, and of the evolution of biological organisms... |
self-reproducing and evolving machines In this paper we explore the work of prominent authors of the nineteenth and early twentieth centuries who addressed this topic. We then identify common themes in their work in terms of the implications of these ideas for the future of human society and evolution, and conclude wit... |
Early writing on self-reproducing and evolving machines Late Nineteenth Century (1860s–1890s) Almost as soon as The Origin of Species was published, some authors began exploring the applicability of Darwin’s ideas to human technology, and the potential consequences that this might entail. |
Samuel Butler: Darwin Among The Machines (1863) and later works As a young man, the English author Samuel Butler (1835–1902) spent five years working in New Zealand. Shortly after his arrival in 1859 he read—and was greatly influenced by—the recently published Origin of Species . During his stay he published a number o... |
Darwin Among the Machines (Butler,, 1863 Butler began the letter by noting the rapid pace of development of machinery from the earliest mechanisms to the most sophisticated examples of the day. He commented that this had far outstripped the pace of development in the animal and vegetable kingdoms, and asked what might ... |
(Butler,, 1863 At that stage, Butler reasoned, the machines would still be reliant upon humans for feeding them, repairing them, and producing their offspring, and hence they would likely treat us kindly. “[Man] will continue to exist, nay even to improve, and will be probably better off in his state of domestication u... |
Throughout his subsequent career, Butler wrestled with his views on the application of Darwin’s theory to machines, and the implications for humanity. In a subsequent letter to The Press entitled Lucubratio Ebria |
(Butler,, 1865 , published on 29 July 1865, he presented a vision whereby machines are seen not as a competing species, but rather as extensions to the human body. From this perspective, Butler emphasised the capacity of machines to exert positive evolutionary influences on the evolution of humankind, not only by incre... |
Upon his return to England in 1864, Butler continued to explore these ideas. They appear in their most developed form in The Book of the Machines , which constituted chapters 23–25 of his novel Erewhon |
(Butler,, 1872 Here he explored the collective reproduction of heterogeneous groups of machines, rather than the reproduction of individuals. Butler likened a complicated machine to “a city or society” (Butler,, 1872 , p. 212) , and asked “how few of the machines are there which have not been produced systematically by... |
(Butler,, 1872 , p. 210) He invoked a number of biological analogies, such as bee pollination and specialisation of reproductive function in ant colonies, to argue that collective machine reproduction is no less like-like than the self-reproduction of individual machines. |
In Erewhon Butler further explored the idea, first addressed in Lucubratio Ebria , that humans and machines are co -evolving, in a process driven by market economics. However, in contrast to his earlier writing, he now feared that this might be detrimental to humankind, with machines evolving by acting parasitically up... |
“For man at present believes that his interest lies in that direction; he spends an incalculable amount of labour and time and thought in making machines breed better and better … and there seem no limits to the results of accumulated improvements if they are allowed to descend with modification from generation to gene... |
Erewhon (Butler,, 1872 , p. 212) As machines evolved to become ever more complex, Butler was concerned that they might “so equalise men’s powers” that evolutionary selection pressure on human physical capabilities would be reduced to a level that precipitated “a degeneracy of the human race, and indeed that the whole b... |
(Butler,, 1872 , p. 224) This concern about the consequences for the human race of entering a long-term co-evolutionary relationship with machines is taken up by a number of later authors, most notably J. D. Bernal, whose work we discuss later. |
Alfred Marshall: Ye Machine (c. 1867) Contemporaneous with Butler, in 1867 the young Alfred Marshall (1842–1924) wrote a series of four papers that formed the basis of talks at “The Grote Club”—an intellectual debating society at the University of Cambridge. His theme was the extent to which the activities of the human... |
The brain of Marshall’s robot consisted of “an indefinite number of wheels of various sizes” connected by bands which would be automatically tightened whenever two wheels moved at the same time |
(Raffaelli,, 1994 , p. 116) . The design therefore implements what would now be classified as a kind of associative learning. He goes on to describe how such a machine might also learn through receiving positive or negative feedback about its actions, and how it might develop instincts to allow it to maintain desired s... |
“Nay, further, the Machine … might make others like itself. We thus get hereditary and accumulated instinct. For these descendants, as they may be called, may vary slightly, owing to accidental circumstances, from the parent. Those which were most suited to the environment would supply themselves most easily with fuel,... |
Alfred Marshall, Ye Machine , c. 1867 (Raffaelli,, 1994 , p. 119) Ye Machine and the other papers presented by Marshall at The Grote Club in the late 1860s had a limited audience at the time, and they were not published in the scientific literature until 1994 (courtesy of the efforts of the late Tiziano Raffaelli). How... |
George Eliot: Impressions of Theophrastus Such (1879) In the following decade, George Eliot (Mary Ann Evans) published her final work, a series of short essays by an imaginary scholar |
(Eliot,, 1879 The chapter Shadows of The Coming Race is a dialogue covering themes first raised by Butler regarding the possibility of machines developing the capacity for self-reproduction and evolution by natural selection. It also touches upon the potential consequences for humans, including mass unemployment and an... |
Early Twentieth Century (1900s–1950s) By the turn of the twentieth century, the pace of technological development had created a more pressing need for considering where such progress might ultimately lead us. During this period, the exploration of potential futures of humanity in a world shared with self-reproducing, e... |
E. M. Forster: The Machine Stops (1909) E. M. Forster’s short story The Machine Stops (Forster,, 1909 was his only work of science fiction. It is now regarded as a classic of dystopian literature (Evans et al.,, 2010 , p. 50) |
The story depicts a future in which humans live underground in personal accommodation where corporeal needs are entirely satisfied by technology (the global, all-nurturing “Machine”). This leaves them free to concentrate on intellectual development, although it also renders them physically degenerate. Forster describes... |
Forster acknowledged the influence of Samuel Butler in his work (Forster,, 1951 —the vision in The Machine Stops of a future where an increasing dependency upon machines leads to the degeneracy of the human body certainly echoes some of Butler’s concerns. Forster’s image of self-maintaining machines sustaining human li... |
Karel Čapek: R.U.R.: Rossum’s Universal Robots (1920) Themes of machine (collective) self-reproduction are further developed in Karel Čapek’s play R.U.R.: Rossum’s Universal Robots |
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